An Approximate Analytical Solution of the Nonlinear Schrödinger Equation with Harmonic Oscillator Using Homotopy Perturbation Method and Laplace-Adomian Decomposition Method

Abstract

The Laplace-Adomian Decomposition Method (LADM) and Homotopy Perturbation Method (HPM) are both utilized in this research in order to obtain an approximate analytical solution to the nonlinear Schrödinger equation with harmonic oscillator. Accordingly, nonlinear Schrödinger equation in both one and two dimensions is provided to illustrate the effects of harmonic oscillator on the behavior of the wave function. The available literature does not provide an exact solution to the problem presented in this paper. Nevertheless, approximate analytical solutions are provided in this paper using LADM and HPM methods, in addition to comparing and analyzing both solutions.

Numerical Optics Soliton Solution of the Nonlinear Schrödinger Equation Using the Laplace and the Modified Laplace Decomposition Method

“…A transformation has been presented so that a system of coupled real partial differential equations is obtained and to be numerically solved in order to approximate the NLSE solution. In spite of, some studies [26,28] in which the Laplace transform has been applied directly to the equation of interest .On the other hand, based on Wazwaz's modification [20] the solution of the NLSE is examined. The obtained results are investigated via illustrations and tables.…”

“…Laplace Decomposition Method (LDM) was introduced by Khuri [11,12] and has been successfully utilized for obtaining solutions of differential equations [6,7,9,14,17,[26][27][28][29][30][31][32][33][34] and the NLSE of our interest. As, for instance, a recent study by Gaxiola [26] who applied the Laplace-Adomian decomposition method to a NLSlike equation, namely the Kundu-Eckhaus equation, and the accuracy as well as the efficiency of the method is proved via examples, as for the nonlinear Schrodinger equation with harmonic oscillator the method of Laplace-Adomian was utilized in a comparison with another semi-analytical method to obtain approximate analytical solutions by Jaradat et al [28] . The Powerfulness of this method is its consistency of Laplace transform and Adomian polynomials which guarantees an accelerative, rapid convergence of series solutions when compared with the ADM itself and therefore provides major progress [11,35,36]…”

Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator

“…Mathematical model of the Problem. The mathematical model represents the oscillatory motion of quantum mechanical oscillator is represented as [26,4,5,36,37,3,29,23,24,25]:…”

“…Results and discussions. for equation (25) by the application of initial condition represented by the equation (26) of ψ (ξ, η). Figure 3 shows that approximate solution obtained by HPSTM has solitary wave nature for one dimensional time fractional nonlinear Schrödinger equation.…”

“…a)-(c) shows the profile of the third order approximation for imaginary part of 1-D wave function for one dimensional timefractional nonlinear Schrödinger equation when k = m = 1 for 0 ≤ η ≤ 0.01 and −5 ≤ ξ ≤ 5 at α = 1, 0.75 and 0.50 for equation(25) by the application of initial condition represented by the equation(26) of ψ (ξ, η).Figure 1shows that approximate solution obtained by HPSTM has solitary wave nature for one dimensional time fractional nonlinear Schrödinger equation.Figure 2 shows the comparison of profile of imaginary part of ψ (ξ, η) at different values of α for (a) −5 ≤ ξ ≤ 5 and η = 0.01 and (b) 0 ≤ η ≤ 1 and ξ = 1.Figure 3 (a)-(c) shows the profile of the third order approximation for real part of 1-D wave function for one dimensional time-fractional nonlinear Schrödinger equation when k = m = 1 for 0 ≤ η ≤ 0.01 and −5 ≤ ξ ≤ 5 at α = 1, 0.75 and 0.50…”